Discrete Mathematical Structures by Pearson

12-46 Discrete Mathematical Structures

or

av cv

jj ij

ik

n

j

k





−=

=+=

∑∑

11

0

since



…



vv v

n12

,,,

are linearly independent, we have

a

1

= a

2

= … = a

k

= c

k

+ 1

= c

k + 2

= … = c

n

= 0

which shows that B is linearly independent.

Hence, dim R(T ) = n - k = dim V - dim N(T ) or dim V = dim R(T ) + dim N(T ).

12.8 NORMED AND INNER PRODUCT SPACES

In this section, we consider only vector spaces over the field of real and complex numbers. The

object is to study vector spaces in which the concepts of ‘length’ or magnitude of a vector and

the angle between two vectors can be defined leading to practical applications.

12.8.1 Norm

The concept of norm provides a way of measuring ...